A Kobayashi–Hitchin correspondence for I±-holomorphic bundles

In this paper, we introduce the notions of α-Hermitian–Einstein metric and α  -stability for I±I±-holomorphic vector bundles on bi-Hermitian manifolds. Moreover, we establish a Kobayashi–Hitchin correspondence for I±I±-holomorphic vector bundles on bi-Hermitian manifolds. Examples of such vector bundles include generalized holomorphic bundles on generalized Kähler manifolds. We also show that α-stability of a vector bundle, in this sense, can depend on the parameter α.